David Cameron's repeated pledges on NHS waiting times are in doubt after the service was warned about its "unacceptable" performance and hospital bosses said targets were likely to be missed.

Hospitals in England have been told to ensure they treat patients within 18 weeks of being referred by their GP and do not leave people waiting in casualty for more than four hours.

David Flory, the NHS deputy chief executive, has ordered the service to do better in order to fulfil the prime minister's recent promises that waiting times would not be allowed to slip.

In a letter to every NHS organisation in England dated 30 June Flory revealed that 59 of them breached the 18-week target in February and March and 32 failed to meet the A&E standard. He reminded NHS bosses that the 18-week pledge is enshrined in the NHS constitution, is a key part of the NHS contract and "the government has recently reinforced this commitment. It is unacceptable for performance to fall below the expected standards as it did in February and March 2011".

In those months 47 hospital trusts failed to treat 90% of admitted patients within 18 weeks while 12 of those also missed the standard for non-admitted patients. The NHS aims to treat 95% of patients within four hours of them arriving at A&E. At 32 trusts that standard was not met.

This disclosure of widespread non-compliance comes as a survey of NHS bosses shows that 53% of chairmen and chief executives of NHS organisations expect patients' access to care – the availability of treatment and waiting times – to worsen over the next year.

The poll, conducted by the NHS Confederation, which represents hospitals, found that 53% of 287 NHS bosses believed that quality in that area of NHS performance would deteriorate. That is an unprecedented finding; access has been improving for several years, though recently began to decline by some key criteria. Some 32% expect the patient's experience of the NHS to worsen too, though 48% think it will improve.

John Appleby, chief economist at The King's Fund health think-tank, said: "The PM's pledge recognised the importance to patients of keeping waiting times short. But with a tight financial situation which is bearing down harder on hospitals than other parts of the NHS, it will be difficult to meet. It is not clear how easing back on central management pressure on hospitals to maintain short waiting times will contribute to meeting the prime minister's pledge."

John Healey, the shadow health secretary, claimed Flory's letter and the poll showed that "this has been a year of broken promises by the prime minister on the NHS and not what people expected when he said he would protect the health service. This is further evidence of widespread anxiety in the health service, as financial pressures impact on patient care and we start to see the NHS going backwards again under the Tories".

Mike Farrar, the NHS Confederation's chief executive, suggested that waiting lists and long waiting times, which have not been seen on a large scale since the NHS Plan in 2000, were making a comeback. There is "a risk" that the 18-week target cannot be maintained, especially as recent quarterly Department of Health data show more people waiting longer than before, Farrar added.

The survey also showed that among NHS bosses:

• 75% expect cuts by local councils to lead to greater demand on hospital, GP and mental health services, especially the cuts to social care.

• Most believe local authority cost-cutting will lead to hospitals seeing more patients having their discharge delayed, more emergency readmissions and more people arriving at A&E

• 42% say the financial pressure on the NHS is "the worst they have ever experienced", while 47% said it was "very serious".

• While 51% thought quality of services would improve, 30% expect it to fall over the next three years

Patient safety could be at risk because the number of managers in the NHS has fallen to "dangerously lose levels", partly as a result of "draconian" cuts and also because bosses have left after suffering regular "denigration" by ministers, including the prime minister, Farrar said. The NHS already has fewer than half the managers seen in France's health system and with a budget of £110bn, the NHS needs a "proper level" of management to do its job properly, he added.

Andrew Lansley, the health secretary, denied waiting times were a problem. "While access to NHS services remains good and waiting times have remained stable, thanks to the hard work of NHS staff, the NHS Confederation has raised concerns about pressures increasing in the future. That is why we have pledged to keep waiting times low and make an extra £2bn available by 2014 to support social care", he said.

Sir Richard Thompson, president of the Royal College of Physicians, which represents hospital doctors, said: "Our Fellows are largely based in hospitals and have very similar concerns about the knock-on effects of the cuts in social care funding in local authorities. This will lead to delayed discharge from hospitals, bed blocking and a detrimental impact on the standard of patient care. The shifting of costs from local authorities to the NHS will create further problems in the long-term."

• This article was amended on 7 July 2011. The original said "32 trusts failed to treat 95% of patients within four hours of them arriving at A&E". This has been clarified.